Designing a teaching situation: Developing formula to calculate the distance from a point to a plane in space (Geometry for the 12th grade, Chapter 3, Lesson 2)

Abstract. In this article, the author designs a teaching situation: developing formula to calculate the distance from a point to a plane in space. In these situations, all learning activities of students will have been planned by the teacher. The formula to calculate the distance from a point to a plane will be created through the process of two different student’s activities: the first is the process of determining the distance in synthetic geometry and the second is the similarity in the formula for calculating the distance from a point to a line in the plane (something that students already know). In this teaching situation, students learn through their own activities but in a manner that was part of the teacher’s plan. In the process of implementing the scenario, from time to time teachers will need to orient students at a minimum level in such a manner that students will find the desired formula.

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JOURNAL OF SCIENCE OF HNUE Interdisciplinary Science, 2013, Vol. 58, No. 5, pp. 47-52 This paper is available online at DESIGNING A TEACHING SITUATION: DEVELOPING FORMULA TO CALCULATE THE DISTANCE FROM A POINT TO A PLANE IN SPACE (Geometry for the 12th grade, Chapter 3, Lesson 2) Bui Van Nghi1 and Nguyen Tien Trung2 1Faculty of Mathematics, Ha Noi National University of Education 2University of Education Publishing House Abstract. In this article, the author designs a teaching situation: developing formula to calculate the distance from a point to a plane in space. In these situations, all learning activities of students will have been planned by the teacher. The formula to calculate the distance from a point to a plane will be created through the process of two different student’s activities: the first is the process of determining the distance in synthetic geometry and the second is the similarity in the formula for calculating the distance from a point to a line in the plane (something that students already know). In this teaching situation, students learn through their own activities but in a manner that was part of the teacher’s plan. In the process of implementing the scenario, from time to time teachers will need to orient students at a minimum level in such a manner that students will find the desired formula. Keywords: Teaching situation, the distance from a point to a plane in space. 1. Introduction When teaching mathematics, particularly geometry, "The teacher needs to create situations in which students must understand the problems, do the work needed to solve problems, adjust his thinking, and attempt to obtain new information.” [2; 93]. In each teaching situation, we believe that the teacher needs to design a structure that contains three basic situations: situation of action, situations of comunication and situations of validation [4]. From the point of view that “doing mathematics properly implies that one is dealing with problems" [5; 22], we need to design a teaching situation where students are given face-to-face situations, and they work on their own and together to solve the problem. At that point students must adjust their thinking to the new information obtained and establish or develop their skills. Received November 05, 2012. Accepted June 25, 2013. Contact Nguyen Tien Trung, e-mail address: trungnt@hnue.edu.vn 47 Bui Van Nghi and Nguyen Tien Trung Therefore, the teaching situation must be designed in such a way that students "will be responsible for the relationship between them and knowledge.” [6; 159] According to a study presented by Bui Van Nghi (2008) [1; 184], in the process of teaching analytic geometry, we "need to pay attention on both the axiomatic method and the method of coordinates." The two methods complement each other, contributing to the improvement of the quality of teaching geometry and the ability to learn it. From theories presented in research, textbooks and teacher’s books, we believe that it is feasible to design geometric teaching situations that are based upon opinions of activities and ideas of the theory of situations. 2. Content In this paper, we present the results of our study: designing the teaching situation to create formula for calculating the distance from a point to a plane in space. The scenario of the teaching situation involves the following actions: * Action 1 (situation of action) The teacher divides the class into four groups and asks each group to solve the following problem: “In space, there is a plane (α) : Ax+By+Cz+D = 0 and there is a pointM (x0; y0; z0). Let’s determine the formula of calculating the distance from a point M to a plane (α). Every student knows how to determine the distance from a point to a plane in space: the distance from a pointM to a plane (α) is equal to the length of segmentMM ′ where M ′ is the perpendicular projection of the pointM in plane (α). Then, students can propose the process (basic process) like this: Step 1: Determine the pointM ′ which is the perpendicular projection of pointM in plane (α); Step 2: Determine the length of segmentMM ′. So, at the moment, all students have the tools: They have gone through the process of calculating this distance previously. However, this process gives them only the segment length (the length of segmentMM ′) and not a formula to determine the length. They have the qualitative methods but not the quantitative method (formula) to arrive at what they are looking for, with hope and faith in the results. Thus, the problem of calculating the distance becomes a problem of calculating the length of a segment or the distance between two points. This kind of problem, for students, is a known problem (in analytic geometry, they have got the formula to solve this problem). So, every student believes that they can do this problem. After a discussion time, each group or individual student can be asked to perform the tasks in the following ways: * Action 2 (situation of action) Option 1: (Use the base process combined with what one knows of vectors in space). Let M ′ (x1; y1; z1) be the perpendicular projection of point M on plane (α), we 48 Designing a teaching situation: developing formula to calculate the distance... have −−−→ MM ′⊥ (α)⇔ −−−→MM ′//~n(α), we have −−−→ MM ′ = t~n(α), (t ∈ R) (2.1) Moreover, we have −−−→ MM ′ = (x1 − x0; y1 − y0; z1 − z0) , ~n(α) = (A;B;C) so the equation (2.1) become this system of equations   x1 − x0 = tA y1 − y0 = tB z1 − z0 = tC ⇔   x1 = x0 + tA y1 = y0 + tB z1 = z0 + tC (2.2) Applying the coordinates of pointM ′ (in the equation (2.2)) in the equation of plane (α), we have A (x0 + tA) +B (y0 + tB) + C (z0 + tC) +D = 0 ⇔ t = −Ax0 +By0 + Cz0 +D A2 +B2 + C2 Then, we have d(M ; (α)) = ∣∣∣−−−→MM ′ ∣∣∣ = √ (At)2 + (Bt)2 + (Ct)2 = √ (A2 +B2 + C2) t2 = |t| √ A2 +B2 + C2 Applying the value of t in the above formula, we have d (M ; (α)) = ∣∣∣∣−Ax0 +By0 + Cz0 +DA2 +B2 + C2 ∣∣∣∣ . √ A2 +B2 + C2 = |Ax0 +By0 + Cz0 +D|√ A2 +B2 + C2 Option 2: (Use the base process combined with knowledge of the vector in space which is presented in the textbook). LetM ′ (x1; y1; z1) be the perpendicular projection of point M on plane (α). We have two vectors −−−→ MM ′ = (x1 − x0; y1 − y0; z1 − z0) and ~n(α) = (A,B,C) which are parallel to each other because they are perpendicular to plane (α). So, we have 49 Bui Van Nghi and Nguyen Tien Trung ∣∣∣−−−→MM ′ ∣∣∣ . ∣∣~n(α)∣∣ = ∣∣∣−−−→MM ′.~n(α) ∣∣∣ = |A (x− x0) +B (y − y0) + C (z − z0)| = |Ax0 + By0 + Cz0 + (−Ax−By − Cz)| (2.3) On the other hand, pointM ′ is in plane (α) so we have Ax1 +By1 + Cz1 +D = 0⇔ D = − (Ax1 +By1 + Cz1) Applying the formula in (1) we have ∣∣∣−−−→MM ′∣∣∣ . ∣∣~n(α)∣∣ = |Ax0 +By0 + Cz0 +D| . Having d (M, (α)) denote the distance from pointM to plane (α), we have d(M, (α)) = ∣∣∣−−−→MM ′ ∣∣∣ = |Ax0 +By0 + Cz0 +D|∣∣~n(α)∣∣ = |Ax0 +By0 + Cz0 +D|√ A2 +B2 + C2 Option 3: (Suggest that students solve the problem looking at the similarity in the formula for calculating the distance from a point to a line in the plane). In the class, students can devise a formula to calculate the distance from a point to a line in the plane. This is therefore a formula to calculate the distance from a point to a plane in space. Suppose that, in a similar way, it has the form d(M, (α)) = |Ax0 +By0 + Cz0 +D|√ A2 +B2 + C2 (2.4) In this case, one can think quickly. The teacher needs to lead his students to test and confirm that which is thought to be true or false. We can do this by the following: 50 Designing a teaching situation: developing formula to calculate the distance... Let M ′ (x1; y1; z1) be the perpendicular projection of point M onto plane (α). We can turn the formula (2.4) into the formula MM ′ = |Ax0 +By0 + Cz0 +D|∣∣~n(α)∣∣ ⇔MM ′. ∣∣~n(α)∣∣ = |Ax0 +By0 + Cz0 +D| ⇔ ∣∣∣−−−→MM ′ ∣∣∣ . ∣∣~n(α)∣∣ = |Ax0 +By0 + Cz0 +D| The angle of the two vectors −−−→ MM ′ = (x1 − x0; y1 − y0; z1 − z0) and ~n(α) = (A,B,C) equals 0 0 or equals 1800. So, we have ∣∣∣−−−→MM ′.~n(α) ∣∣∣ = ∣∣∣ ∣∣∣−−−→MM ′ ∣∣∣ . ∣∣~n(α)∣∣ cos (−−−→ MM ′, ~n(α) )∣∣∣ = ∣∣∣−−−→MM ′∣∣∣ . ∣∣~n(α)∣∣ Inferred ∣∣∣−−−→MM ′ ∣∣∣ . ∣∣~n(α)∣∣ = ∣∣∣−−−→MM ′.~n(α) ∣∣∣ = |A (x1 − x0) +B (y1 − y0) + C (z1 − z0)| = |Ax0 +By0 + Cz0 + (−Ax1 − By1 − Cz1)| (2.5) On the other hand, pointM ′ belongs to plane (α) and we have Ax1 +By1 + Cz1 +D = 0⇔ D = − (Ax1 +By1 + Cz1) Applying the above formula (2.5), we have ∣∣∣−−−→MM ′ ∣∣∣ . ∣∣~n(α)∣∣ = |Ax0 +By0 + Cz0 +D| . So, the previous judgement is seen to be correct. * Action 3: Validation and applying the formula to exercises (situation of action and validation) Exercise 1. Calculate the distance from point to plane (α) in space for each of the following cases: a)M (1;−2; 13) , (α) : 2x− 2y − z + 3 = 0 b)M (1;−2; 3) , (α) : x− 5y − 2z + 5 = 0 c)M (1; 2; 2) , (α) : 3x− y − z + 1 = 0 51 Bui Van Nghi and Nguyen Tien Trung d)M (−3; 1; 5) , (α) : z = 0 Exercise 2. Calculate the distance between two parallel planes (P ), (Q): (P ) : x+ 2y + 2z + 11 = 0; (Q) : x+ 2y + 2z + 2 = 0 Exercise 3. a) Let the two planes (α) : x−2y+2z−11 = 0 and (β) be two parallel planes. Find the equation of plane (β) if the distance between planes (α) and (β) equals 3. b) Find pointM which is on line Oz such that the two planes having the equations: x+ y − z + 1 = 0; x− y + z + 5 = 0 are equidistant. * Action 4: Legitimization and institutionalization (situation of validation) The teacher leads his students to come to the conclusion: In space Oxyz, the distance from point M (x0; y0; z0) to the plane (α) : Ax + By + Cz + D = 0 equals d(M, (α)) = |Ax0 +By0 + Cz0 +D|√ A2 +B2 + C2 . The teacher also needs to let his students experience this process: when they face cognitive obstacles, they can look at how they can simplify it and look for aspects that are already known. So, the two formulas of distance (the distance from a point to a line in the plane and the distance from a point to a plane in space) have similar forms. 3. Conclusion In the teaching scenario presented above, students play a role in the discovery in an encouraging and interactive way. In the process of implementing such a scenario, teachers can periodically provide support to orient students at a minimum level to allow students to find the desired formula. REFERENCES [1] Bui Van Nghi, 2008. Methodology of teaching Mathematics in detail. University of Education Publishing House, Hanoi. [2] Bui Van Nghi, 2008. Applying theory of teaching to teach Mathematics. University of Education Publishing House, Hanoi. [3] Geometry Text books for the 12th Grade, Teacher’s books, Geometry Student Exercise books for the 12th Grade (basic and advance), Vietnam Education Publishing House, Hanoi. [4] Virginia M. Warfield, 2006. Invitation to Didactique. University of Washington, Seattle, Washington. [5] Guy Brousseau, 2002. Theory of Didactical situations in mathematics. Vol. 19, Kluwer Academic Publishers. [6] Annie Bessot, Claude Comiti, Le Thi Hoai Chau, Le Van Tien, 2009. Elements of Theory of Didactical situations in Mathematics. Bilingual book Vietnamese - French, Ho Chi Minh National University of Education Publishing House. 52
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